Piecewise-deterministic Markov process: Difference between revisions

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In [[probability theory]], a '''piecewise-deterministic Markov process (PDMP)''' is a process whose behaviour is governed by random jumps at points in time, but whose evolution is deterministically governed by an [[ordinary differential equation]] between those times. The class of models is "wide enough to include as special cases virtually all the non-diffusion models of [[applied probability]]."<ref name="davis" /> The process is defined by three quantities: the flow, the jump rate, and the transition measure.<ref name="siam2010">{{Cite journal | last1 = Costa | first1 = O. L. V. | last2 = Dufour | first2 = F. | doi = 10.1137/080718541 | title = Average Continuous Control of Piecewise Deterministic Markov Processes | journal = SIAM Journal on Control and Optimization | volume = 48 | issue = 7 | pages = 4262 | year = 2010 | arxiv = 0809.0477| s2cid = 14257280 }}</ref>

The model was first introduced in a paper by [[Mark H. A. Davis]] in 1984.<ref name="davis">{{Cite journal | last1 = Davis | first1 = M. H. A. | author-link = Mark H. A. Davis| title = Piecewise-Deterministic Markov Processes: A General Class of Non-Diffusion Stochastic Models | journal = Journal of the Royal Statistical Society. Series B (Methodological)| volume = 46 | issue = 3 | pages = 353–388 | doi = 10.1111/j.2517-6161.1984.tb01308.x| jstor = 2345677| year = 1984 }}</ref>

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Löpker and Palmowski have shown conditions under which a [[reversed process|time reversed]] PDMP is a PDMP.<ref>{{Cite journal | last1 = Löpker | first1 = A. | last2 = Palmowski | first2 = Z. | doi = 10.1214/EJP.v18-1958 | title = On time reversal of piecewise deterministic Markov processes | journal = Electronic Journal of Probability | volume = 18 | year = 2013 | arxiv = 1110.3813| s2cid = 1453859 }}</ref> General conditions are known for PDMPs to be stable.<ref>{{Cite journal | last1 = Costa | first1 = O. L. V. | last2 = Dufour | first2 = F. | doi = 10.1137/060670109 | title = Stability and Ergodicity of Piecewise Deterministic Markov Processes | journal = SIAM Journal on Control and Optimization | volume = 47 | issue = 2 | pages = 1053 | year = 2008 | url = http://www.producao.usp.br/bitstream/BDPI/14708/1/art_COSTA_Stability_and_ergodicity_of_piecewise_deterministic_Markov_2008.pdf }}</ref>

Galtier andet Alal.<ref>{{Cite journal | last1 = Galtier | first1 = T. | doi = 10.1051/ps/2019015 | title =On the optimal importance process for piecewise deterministic Markov process | journal = ESAIMEsaim: PSPs | volume = 23 | year = 2019 | pages = 893–921 | s2cid = 198467101 | doi-access = free }}</ref> studied the law of the trajectories of PDMP and provided a reference measure in order to express a '''density of a trajectory''' of the PDMP. Their work opens the way to any application using densities of trajectory. (For instance, they used the density of a trajectories to perform [[importance sampling]], this work was further developed by Chennetier and Al.<ref>{{cite arXiv| last1 = Chennetier | first1 = G. | title =Adaptive importance sampling based on fault tree analysis for piecewise deterministic Markov process | year = 2022 | class = stat.CO | eprint = 2210.16185 }}</ref> to estimate the reliability of industrial systems.)

==See also==

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[[Category:Markov processes]]

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